Online gaming is a complicated issue. The Chromebooks run a full version of Chrome so if it is a flash-based game, it should work just fine. Some online gaming sites, though, use windows or mac programs as part of their site and ChromeOS can't handle those. Your best bet is to google "chromebook" and "the site(s) that you like" and i'm sure there will be a discussion, somewhere, about people being happy or upset with their experience. I don't play any online games, other than the ones available in the Chrome Store, so I can't give you any real feedback.
I am not sure of your level of gaming when you ask? My son is 10 1/2 and plays lots of free internet games and has no problems? But if your heavy serious gamer--it will not have the high end graphics and spectacular sounds that some more top shelf laptops have. We love it!!
It is very unlikely that the ARM Chromebook would play compatible games better than a typical laptop - unless you're talking about an older one running Windows XP or Vista. You might get equivalent performance to the cheapest laptops, and it should be better than netbooks. However, if you want a high level of performance, you might need the Chromebook Pixel.